Stochastic Stackelberg differential games modeling crowd-authority interaction dynamics

This paper studies a Stackelberg differential game for crowd dynamics control which is performed by an authority (or multiple authorities). The problem consists of the optimal design of policies in the game-theoretical framework of hierarchical mechanism design. We present a two-layer game where there is a strategic selection, made by a single or multiple authorities, that affects the behavior of the crowd dynamics by means of either an attractive force or the simultaneous combination of attractive and repulsive forces. The crowd reacts against the imposed policies, and the authority optimally designs these policies knowing that the crowd will perform a best-response strategy against them. For each case, comprising the consideration of only an attractive force and the combination of attractive and repulsive forces, we present the resulting crowd dynamics in terms of Hamilton–Jacobi–Bellman (HJB) and Fokker–Planck–Kolmogorov (FPK) Partial Differential Equations (PDEs), a Hughes-like crowd modeling, the corresponding Stackelberg equilibrium for the crowd-authority interaction, and linear quadratic examples for which semi-explicit solution can be computed by freezing the crowd at different time steps. We present two different algorithms that can be used to solve the crowd dynamics model. The first algorithm proposes to freeze and update the crowd (density) in different time steps. The second algorithm is an improvement over the first one, and it proposes to use the backward solution of the value function, initially obtained by freezing the crowd, to estimate the propagation of the crowd (density) in an iterative fashion. Finally, we develop some simulations to illustrate the results presented in this paper.